This talk is devoted to differential operators on metric graphs, also called quantum graphs. Such operators are induced by metric graphs with edges equipped by differential expressions and with matching conditions at graph vertices. We suppose that the operator coefficients (e.g., Sturm-Liouville potentials) are known a priori on some edges of the graph, and we need to recover the operator coefficients on the remaining part of the edges from some spectral data. The problems of this kind are called the partial inverse problems. Various types of partial inverse problems will be discussed for special types of graphs (star-shaped graphs, simple graphs with cycles) and also for graphs of arbitrary geometrical structure.